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Topic: Stochastic matrix

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Hermitian matrix

Symmetric matrix

Transition matrix

Matrix (mathematics)

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Examples of Markov chains

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	PlanetMath: permutation matrix
		Since they have a single 1 in each row and each column, they are doubly stochastic.
		They are the extreme points of the convex set of doubly stochastic matrices.
		This is version 13 of permutation matrix, born on 2002-01-04, modified 2007-08-23.
	planetmath.org /encyclopedia/PermutationMatrix.html (125 words)

	Stochastic matrix - Wikipedia, the free encyclopedia
		In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a left stochastic matrix is a square matrix whose columns are probability vectors, i.e., the entries in each column are nonnegative real numbers whose sum is 1.
		Likewise, a right stochastic matrix is a square matrix whose rows are probability vectors.
		In a doubly stochastic matrix all rows and all columns are probability vectors.
	en.wikipedia.org /wiki/Stochastic_matrix (217 words)

	Stochastic - Wikipedia, the free encyclopedia
		Stochastic, from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random.
		A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is not fully determined by the previous state of the environment.
		A stochastic matrix is a matrix that has non-negative real entries that sum to 1 in each column.
	en.wikipedia.org /wiki/Stochastic (674 words)

	Stochastic Matrix -- from Wolfram MathWorld
		A stochastic matrix is the transition matrix for a finite Markov chain, also called a Markov matrix.
		Elements of the matrix must be real numbers in the closed interval [0, 1].
		A completely independent type of stochastic matrix is defined as a square matrix with entries in a field
	mathworld.wolfram.com /StochasticMatrix.html (123 words)

	Matrix Tutorial 1: Stochastic Matrices
		The sum of the principal is the trace of the matrix.
		Accordingly, the rank of a square matrix is equal to the number of nonzero rows in its upper-triangular matrix or the number of nonzero columns in its equivalent lower-triangular matrix, whichever of these two number is smaller.
		In general, a matrix whose sum of all row elements (or column elements) equals 1 is called a stochastic matrix.
	www.miislita.com /information-retrieval-tutorial/matrix-tutorial-1-stochastic-matrices.html (2152 words)

	Abstract of: A stochastic uncoupling process for graphs
		The process takes a stochastic matrix as input, and then alternates flow expansion and flow inflation, each step defining a stochastic matrix in terms of the previous one.
		It is shown that for r∈ N and for M a stochastic dpsd matrix, the image Γr M is again dpsd.
		Determinantal inequalities satisfied by a dpsd matrix M imply a natural ordering among the diagonal elements of M, generalizing a mapping of nonnegative column allowable idempotent matrices onto overlapping clusterings.
	db.cwi.nl /rapporten/abstract.php?abstractnr=615 (336 words)

	PlanetMath: Google matrix
		Google's PageRank algorithm uses a particular stochastic matrix called the Google matrix.
		The purpose of the PageRank algorithm is to compute a stationary vector of the Google matrix.
		This is version 3 of Google matrix, born on 2007-04-29, modified 2007-04-29.
	planetmath.org /encyclopedia/GoogleMatrix.html (112 words)

	Citations: A relationship between arbitrary positive matrices and doubly stochastic matrices - Sinkhorn (ResearchIndex)
		First, the matrix W is set to zero; entries for matches satisfying the geometric constraints of Section 4.2.2, as well as all entries in row M 1 and column N 1, are then assigned an initial value of one.
		First, the matrix W is set to zero; entries for matches satisfying the geometric constraints of 3.2.2, as well as all entries in row M 1 and column N 1, are then assigned an initial value of....
		When each row and column of a square correspondence matrix is normalized (several times, alternatingly) by the sum of the elements of that row or column respectively, the resulting matrix has positive elements with all rows and columns summing to one.
	citeseer.ist.psu.edu /context/229348/0 (2204 words)

	Stochastic Processes and Queuing Models, Queueing Theory - Numericana
		The becquerel may be loosely described as the "stochastic reciprocal of the second" (the herz is another "reciprocal of the second" which is used for periodic phenomena, not random ones).
		For such a process, a generator matrix Q is defined (also called a transition rate matrix) as the time derivative of the stochastic matrix that gives the probability of ending up in state j at time t, starting from state i at time 0.
		The very definition of the activity of a stochastic process makes it clear that when several Poisson processes are so combined, the resulting process is a process whose activity is the sum of the activities of the component processes.
	home.att.net /~numericana/answer/stochastic.htm (1774 words)

	an introduction to the MCL algorithm
		For stochastic matrices diagonally similar to a symmetric matrix, the type of limit invariably found is that of a doubly idempotent matrix; idempotent under both matrix squaring and matrix inflation.
		Interpreting the connected components as clustering yields a clustering whose distribution is invariably strongly related to the density characteristics of the input matrix.
		The clustering associated with such a matrix is stable under perturbations of the MCL process (that is, it is essentially defined by the block structure), except for the phenomenon of overlap.
	micans.org /mcl/intro2.html (731 words)

	Markov Chains 1
		If we do this for every pitch value we build a stochastic matrix which resembles the composition that it is based on.
		It is the 'seed' that is the difference between a normal stochastic matrix and a Markov matrix.
		Remember from the tutorial on stochastic matrices that the choice of output is based on the random number being smaller than the sum of weighings going left to right across the row.
	jmusic.ci.qut.edu.au /jmtutorial/Markov1.html (679 words)

	an introduction to the MCL algorithm
		Inflation corresponds with taking the Hadamard power of a matrix (taking powers entrywise), followed by a scaling step, such that the resulting matrix is stochastic again, i.e.
		A column stochastic matrix is a non-negative matrix with the property that each of its columns sums to 1.
		Each column j of a stochastic matrix M corresponds with node j of the stochastic graph associated with M.
	micans.org /mcl/intro.html (1187 words)

	MarkovClustering
		MarkovClustering implements the Markov clustering (MCL) algorithm for graphs, using a HashMap-based sparse representation of a Markov matrix, i.e., an adjacency matrix m that is normalised to one.
		The algorithm starts by creating a Markov matrix from the graph, for which first the adjacency matrix is added diagonal elements to include self-loops for all nodes, i.e., probabilities that the random walker stays at a particular node.
		The expansion step corresponds to matrix multiplication (on stochastic matrices), the inflation step corresponds with a parametrized inflation operator Gamma_r, which acts column-wise on (column) stochastic matrices (here, we use row-wise operation, which is analogous).
	www.arbylon.net /projects/knowceans-mcl/doc/org/knowceans/mcl/MarkovClustering.html (611 words)

	Stochastic Matrix
		This is an important detail about stochastic matrices, the sum of all weighted choices must be equal to 1.0 and therefore if any individual choice is weighted to 1.0 all other choices must be equal to 0.0.
		Another reason for the popularity of stochastic processes is the amount of data which they produce.
		A stochastic matrix is capable of generating any number of outcomes.
	jmusic.ci.qut.edu.au /jmtutorial/Markov0.html (281 words)

	Averaging and convergence
		The fact that the largest eigenvalue of a positive matrix is isolated means that powers of the matrix should converge to a multiple of the idempotent matrix formed from the corresponding eigenvectors.
		Conversely, one way of estimating the second eigenvalue is to study the behaviour of matrix powers, preferably already having transformed the matrix to stochastic from.
		Suppose that M is the matrix, that X is a real vector whose algebraically largest and smallest components are A and a respectively, and that the corresponding extreme elements of MX are B and b, respectively.
	delta.cs.cinvestav.mx /~mcintosh/oldweb/lcau/node101.html (724 words)

	How Google Finds Your Needle in the Web's Haystack
		This has the effect of modifying the hyperlink matrix H by replacing the column of zeroes corresponding to a dangling node with a column in which each entry is 1/n.
		The matrix S has the pleasant property that the entries are nonnegative and the sum of the entries in each column is one.
		To summarize, the matrix S is stochastic, which implies that it has a stationary vector.
	www.ams.org /featurecolumn/archive/pagerank.html (3061 words)

	The Stochastic Subspace Identification Techniques
		One of the typical parametric model structures to use in output-only modal analysis of linear and time-invariant physical systems is the stochastic state space system.
		So the fundamental problem to solve in the stochastic subspace identification technique is to extract the predicted states from the measured data.
		It is actually the estimation of this matrix that can be done in different ways and results in that several stochastic subspace identification techniques exist.
	www.svibs.com /literature/Notes/Modal_Analysis/Note_on_SSI.htm (1768 words)

	Probabilistic de Bruijn matrix
		Any matrix intended to preserve each and every such column vector must necessarily have columns of unit sum, as can be seen by testing the matrix on unit vectors.
		Expressing either constraint in matrix form guarantees a unit eigenvalue, with a probability vector as the equilibrium eigenvector of opposite handedness.
		These and certain other conclusions characterize stochastic matrices, all of which information can be found in any one of several recent textbooks[41,13,103] which can be consulted for descriptions and proofs of the results.
	delta.cs.cinvestav.mx /~mcintosh/oldweb/lcau/node74.html (670 words)

	Matrix Set STOCH
		This stochastic matrix is derived from the application of Markov modeling techniques to the analysis and evaluation of computer systems.
		The matrixis a 163 by 163 stochastic matrix with 935 nonzero entries.
		The Matrix Market is a service of the Mathematical and Computational Sciences Division / Information Technology Laboratory / National Institute of Standards and Technology.
	math.nist.gov /MatrixMarket/data/NEP/stoch/stoch.html (168 words)

	Amazon.com: Introduction to Stochastic Processes: Books: Gregory F. Lawler (Site not responding. Last check: 2007-10-26)
		Well-chosen examples and interesting exercises make this text a good choice for a first course in stochastic processes for a broad class of students.
		This concise, informal introduction to stochastic processes evolving with time was designed to meet the needs of graduate students not only in mathematics and statistics, but in the many fields in which the concepts presented are important, including computer science, economics, business, biological science, psychology, and engineering.
		A stochastic process is a random process evolving with time.
	www.amazon.com /Introduction-Stochastic-Processes-Gregory-Lawler/dp/0412995115 (1013 words)

	ABSTRACT: Stochastic Matrix Structure (Site not responding. Last check: 2007-10-26)
		On The Structure Of Stochastic Matrices With A Subdominant Eigenvalue Near 1
		An nxn irreducible stochastic matrix P can possess a subdominant eigenvalue near 1.
		In this article we clarify the relationship between the nearness of these eigenvalues and the nearly uncoupling (some authors say ``nearly completely decomposable'') of P. We prove that for fixed n, if the subdominant eigenvalue is sufficiently close to 1, then P is nearly uncoupled.
	meyer.math.ncsu.edu /Meyer/Abstracts/StochasticMatrixStructure.html (126 words)

	Markov Chains
		Definition 3.1 A stochastic matrix is a matrix with nonnegative entries and row sums of
		A stochastic matrix is ergodic if it (precisely, the digraph of its nonzero entries) is strongly connected and aperiodic.
		A walk on a Cayley graph is always doubly stochastic.
	people.cs.uchicago.edu /~laci/reu03/n2_12/node3.html (215 words)

	SPN2MGM: Tool Support for Matrix-Geometric Stochastic Petri Nets - MGM, for, stochastic, In, the, Computer, ...
		Abstract: In this paper we present the tool spn2mgm that can be used to construct and solve stochastic Petri nets using matrix-geometric techniques.
		The tool automatically recognizes the "matrix-geometric structure" of the Markov chain underlying the stochastic Petri net, and solves the Markov chain with these well-known and efficient techniques.
		We informally characterize the class of stochastic Petri nets of interest (a formal definition has been given in an earlier paper) after which we briefly touch...
	citeseer.ist.psu.edu /218855.html (857 words)

	Markov Chains
		A matrix for which all the column vectors are probability vectors is called transition or stochastic matrix.
		Note that the matrix A is a stochastic matrix!
		In fact there is a general result similar to the one above for any stochastic matrix.
	www.sosmath.com /matrix/markov/markov.html (491 words)

	[No title]
		Starts by creating a random 0/1 valued matrix, then normalizes the rows and columns.
		One consists of pointers to floats, the remainder of floats.] ****************************************************************************/ randomDSSMatrix::randomDSSMatrix(int n) { dimension = n; matrix** = new (double*)[n]; for(int i=0; i
		This is done by computing perfect matchings (actually maximum matchings which we know must be perfect, thanks to Hall's theorem), and then pulling out the corresponding permutation matrix.
	www.ece.utexas.edu /~adnan/network-02/bvn.cc (425 words)

	Doubly Stochastic Matrix -- from Wolfram MathWorld
		In other words, both the matrix itself and its transpose are stochastic.
		Horn, A. "Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix." Amer.
		Sherman, S. "Doubly Stochastic Matrices and Complex Vector Spaces." Amer.
	mathworld.wolfram.com /DoublyStochasticMatrix.html (201 words)

	m0000002.htm
		We explore the use of the matrix exponential as defined by (3) in modeling
		, all of which are proportional to a row-stochastic matrix.
		The row-stochastic weight matrix has very favorable numeric as well as statistical properties.
	www.spatial-statistics.com /spatial_statistical_manuscripts/MESS/html/m0000002.htm (444 words)

	Citebase - Unitary stochastic matrix ensembles and spectral statistics (Site not responding. Last check: 2007-10-26)
		We propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with Markov processes on graphs.
		We argue that the spectral statistics of such an ensemble (after ensemble averaging) depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix.
		The set of bistochastic or doubly stochastic N by N matrices form a convex set called Birkhoff's polytope, that we describe in some detail.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0104014 (1206 words)

	SPN2MGM: Tool Support for Matrix-Geometric Stochastic Petri Nets (Site not responding. Last check: 2007-10-26)
		In this paper we present the tool SPN2MGM that can be used to construct and solve stochastic Petri nets using matrix-geometric techniques.
		We informally characterize the class of stochastic Petri nets of interest (a formal definition has been given in an earlier paper) after which we briefly touch upon the matrix-geometric solution approach.
		In particular, we present a model of a queueing system in which check- pointing can be used to shorten the recovery process after server- breakdowns have occurred.
	csdl.computer.org /comp/proceedings/ipds/1996/7484/00/74840219abs.htm (232 words)

	IngentaConnect Stochastic matrix description of glass transition in ternary chal...
		IngentaConnect Stochastic matrix description of glass transition in ternary chal...
		Stochastic matrix description of glass transition in ternary chalcogenide systems
		You may be required to register, activate a subscription or purchase the article before you can obtain the full text.
	www.ingentaconnect.com /content/els/00223093/1998/00000231/00000001/art00417 (117 words)

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